Poisson-Boltzmann equation derivation

The modified Poisson-Boltzmann equation derived by Camie and McLaughhn [5] for the above system then reads... 

Tanford, C., Kirkwood, J. G. Theory of protein titration curves. I. General equations for impenetrable spheres. J. Am. Chem. Soc. 79 (1957) 5333-5339. 6. Garrett, A. J. M., Poladian, L. Refined derivation, exact solutions, and singular limits of the Poisson-Boltzmann equation. Ann. Phys. 188 (1988) 386-435. Sharp, K. A., Honig, B. Electrostatic interactions in macromolecules. Theory and applications. Ann. Rev. Biophys. Chem. 19 (1990) 301-332. 

Here, b is the distance between the nearest unit charges along the cylinder (b = 0.34nm for the ssDNA and b = 0.17nm for the dsDNA), (+) and (—) are related to cations and anions, respectively, and a = rss for the ssDNA and a rds for the dsDNA. The expressions (5) and (6) have been obtained using the equations for the electrostatic potential derived in [64, 65], where a linearization of the Poisson-Boltzmann equation near the Donnan potential in the hexagonal DNA cell was implemented. 

Derive and solve the appropriate linear Poisson-Boltzmann equation for the interface between two immiscible solutions. 

The purpose of the present chapter is to introduce some of the basic concepts essential for understanding electrostatic and electrical double-layer pheneomena that are important in problems such as the protein/ion-exchange surface pictured above. The scope of the chapter is of course considerably limited, and we restrict it to concepts such as the nature of surface charges in simple systems, the structure of the resulting electrical double layer, the derivation of the Poisson-Boltzmann equation for electrostatic potential distribution in the double layer and some of its approximate solutions, and the electrostatic interaction forces for simple geometric situations. Nonetheless, these concepts lay the foundation on which the edifice needed for more complicated problems is built. 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

Using Poisson-Boltzmann theory we can derive a simple expression for the disjoining pressure. For the linear case (low potentials) and for a monovalent salt, the one-dimensional Poisson-Boltzmann equation (Eq. 4.9) is... 

In this Appendix, equations will be derived for the double layer interaction between two charged, planar surfaces in an electrolyte-free system. W e assume that the potential ip(x) obeys the Poisson—Boltzmann equation... 

In the paragraphs below, we first examine the simple, analytical results that can be derived from the linear Poission-Boltzmann equation for a single particle interacting with a flat surface. Next, more complicated physical situations are considered, including interactions between many particles and a wall between a particle and a deformable interface between a protein and a wall and between a moving particle and a wall. In Sec. Ill, solutions to the nonlinear Poisson-Boltzmann equation are considered, and comparisons are made between the linear and nonlinear versions and also with more... 

This simple and appealing result shows that, for H 1 /k, the sphere-wall interaction depends linearly on the charge densities of each surface, and decays exponentially with the separation distance. The result does not depend on whether the surfaces are considered to be constant charge density or constant potential, because the potentials of an isolated wall and sphere were used in its derivation. Phillips [13] has compared Eq. (24) with a numerical solution of the linear Poisson-Boltzmann equation, and shows that it errs by less than about 10% for xh>3 when 0.5 [Pg.257]

Unlike the other examples in this section, the equation governing the electrostatics here [i.e., Eq. (53)] is not the linearized Poisson-Boltzmann equation. However, considering interactions outside of thin double layers does have the effect of linearizing the problem. In Eq. (54), n is the fluid viscosity, K is the conductivity, is the zeta potential of the z th surface, and is a bipolar coordinate that is constant on the sphere and wall surfaces. It is this last condition (54), derived by Bike and Prieve [36] as a requirement to satisfy charge conservation, that couples the fluid mechanics with the electrostatics. 

In all of the discussion above, comparisons have been made between various types of approximations, with the nonlinear Poisson-Boltzmann equation providing the standard with which to judge their validity. However, as already noted, the nonlinear Poisson-Boltzmann equation itself entails numerous approximations. In the language of liquid state theory, the Poisson-Boltzmann equation is a mean-field approximation in which all correlation between point ions in solution is neglected, and indeed the Poisson-Boltzmann results for sphere-sphere [48] and plate-plate [8,49] interactions have been derived as limiting cases of more rigorous approaches. For many years, researchers have examined the accuracy of the Poisson-Boltzmann theory using statistical mechanical methods, and it is... 

Without going into much mathematical detail, the capacity of the space charge region is derived by solving the Poisson-Boltzmann equation as... 

Poisson-Boltzmann equation — The Poisson-Boltz-mann equation is a nonlinear, elliptic, second-order, partial differential equation which plays a central role, e.g., in the Gouy-Chapman (- Gouy, - Chapman) electrical -> double layer model and in the - Debye-Huckel theory of electrolyte solutions. It is derived from the classical -> Poisson equation for the electrostatic potential... 

Because this result has been obtained by solving a generalized Poisson-Boltzmann equation with the linearization approximation, it is necessary to compare it with the DLVO theory in the limit where the Debye approximation holds. In this case, Verwey and Overbeek [2], working in cgs (centimeter-gram-second) units, derived the following approximate equation for the repulsive potential ... 

Fig. 3.36. Steps in the derivation of the linearized Poisson-Boltzmann equation.

This is the Poisson-Boltzmann equation for the potential distribution i/ (r). The surface charge density a of the particle is related to the potential derivative normal to the particle surface as... 

When the magnimde of the surface potential is arbitrary so that the Debye-Hiickel hnearization cannot be allowed, we have to solve the original nonlinear spherical Poisson-Boltzmann equation (1.68). This equation has not been solved but its approximate analytic solutions have been derived [5-8]. Consider a sphere of radius a with a... 

By using an approximation method similar to the above method and the method of White [6], one can derive an accurate analytic expression for the potential distribution around a spherical particle. Consider a sphere of radius u in a symmetrical electrolyte solution of valence z and bulk concentration n[7]. The spherical Poisson-Boltzmann equation (1.68) in this case becomes... 

So far we have treated uniformly charged planar, spherical, or cylindrical particles. For general cases other than the above examples, it is not easy to solve analytically the Poisson-Boltzmann equation (1.5). In the following, we give an example in which one can derive approximate solutions. 

We give below a simple method to derive an approximate solution to the hnear-ized Poisson-Boltzmann equation (1.9) for the potential distribution i/ (r) around a nearly spherical spheroidal particle immersed in an electrolyte solution [12]. This method is based on Maxwell s method [13] to derive an approximate solution to the Laplace equation for the potential distribution around a nearly spherical particle. 

The Poisson-Boltzmann equation for the potential distribution around a cylindrical particle without recourse to the above two assumptions for the limiting case of completely salt-free suspensions containing only particles and their counterions was solved analytically by Fuoss et al. [1] and Afrey et al. [2]. As for a spherical particle, although the exact analytic solution was not derived, Imai and Oosawa [3,4] smdied the analytic properties of the Poisson-Boltzmann equation for dilute particle suspensions. The Poisson-Boltzmann equation for a salt-free suspension has recently been numerically solved [5-8]. 

In this section, we present a novel linearization method for simplifying the nonlinear Poisson-Boltzmann equation to derive an accurate analytic expression for the interaction energy between two parallel similar plates in a symmetrical electrolyte solution [13, 14]. This method is different from the usual linearization method (i.e., the Debye-Hiickel linearization approximation) in that the Poisson-Boltzmann equation in this method is linearized with respect to the deviation of the electric potential from the surface potential so that this approximation is good for small particle separations, while in the usual method, linearization is made with respect to the potential itself so that this approximation is good for low potentials. 

This chapter deals with a method for obtaining the exact solution to the linearized Poisson-Boltzmann equation on the basis of Schwartz s method [1] without recourse to Derjaguin s approximation [2]. Then we apply this method to derive series expansion representations for the double-layer interaction between spheres [3-13] and those between two parallel cylinders [14, 15]. 

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